- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 03:34:36
19
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=20&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
20
2018
1
Fractional elliptic quasi-variational inequalities: Theory and numerics
Harbir
Antil
George Mason University, Fairfax, USA
Carlos
Rautenberg
Humboldt-Universität zu Berlin, Germany
Quasivariational inequality, QVI, fractional derivatives, fractional diffusion, free boundary problem, Caffarelli–Silvestre and Stinga–Torrea extension, weighted Sobolev spaces, Mosco convergence, fixed point algorithm, finite element method
This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order $s \in (0, 1)$, studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.
Partial differential equations
Real functions
Calculus of variations and optimal control; optimization
Numerical analysis
1
24
10.4171/IFB/395
http://www.ems-ph.org/doi/10.4171/IFB/395
5
3
2018
Approximation and characterization of quasi-static $H^1$-evolutions for a cohesive interface with different loading-unloading regimes
Matteo
Negri
Università di Pavia, Italy
Enrico
Vitali
Università di Pavia, Italy
Cohesive fracture, quasi-static propagation, BV-evolution
We consider the quasi-static evolution of a prescribed cohesive interface: dissipative under loading and elastic under unloading. We provide existence in terms of parametrized $BV$-evolutions, employing a discrete scheme based on local minimization, with respect to the $H^1$-norm, of a regularized energy. Technically, the evolution is fully characterized by: equilibrium, energy balance and Karush–Kuhn–Tucker conditions for the internal variable. Catastrophic regimes (discontinuities in time) are described by gradient flows of visco-elastic type.
Calculus of variations and optimal control; optimization
Mechanics of deformable solids
25
67
10.4171/IFB/396
http://www.ems-ph.org/doi/10.4171/IFB/396
5
3
2018
On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers
Matthieu
Bonnivard
Université Denis Diderot – Paris 7, France
Antoine
Lemenant
Université Paris Diderot – Paris 7, France
Vincent
Millot
Université Denis Diderot - Paris 7, France
Steiner problem, gamma-convergence, Ginzburg–Landau, Modica–Mortola, phase field approximation
In this article, we consider and analyse a variant of a functional originally introduced in [9, 27] to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter $\epsilon > 0$ and resembles the (scalar) Ginzburg–Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as $\epsilon \to 0$, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.
Calculus of variations and optimal control; optimization
69
106
10.4171/IFB/397
http://www.ems-ph.org/doi/10.4171/IFB/397
5
3
2018
The Verigin problem with and without phase transition
Jan
Prüss
Martin-Luther-Universität Halle-Wittenberg, Germany
Gieri
Simonett
Vanderbilt University, Nashville, USA
Two-phase flows, phase transition, Darcy’s law, Forchheimer’s law, available energy, quasilinear parabolic evolution equations, maximal regularity, generalized principle of linearized stability, convergence to equilibria
Isothermal incompressible two-phase flows with and without phase transition are modeled, employing Darcy’s and/or Forchheimer’s law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria are identified and their thermodynamical stability is investigated by means of a variational approach. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. It is further shown that a non-degenerate equilibrium is dynamically stable in the natural state manifold if and only if it is thermodynamically stable. Finally, it is shown that a solution which does not develop singularities exists globally and converges to an equilibrium in the state manifold.
Partial differential equations
Fluid mechanics
107
128
10.4171/IFB/398
http://www.ems-ph.org/doi/10.4171/IFB/398
5
3
2018
Minimal partitions for $p$-norms of eigenvalues
Beniamin
Bogosel
École Normale Supérieure, Paris, France
Virginie
Bonnaillie-Noël
École Normale Supérieure, Paris, France
Minimal partitions, shape optimization, Dirichlet–Laplacian eigenvalues, numerical simulations
In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a $p$-norm of eigenvalues of the Dirichlet–Laplace operator. The extremal case of the infinity norm, where we minimize the largest fundamental eigenvalue of each cell, is one of our main interests. We propose three numerical algorithms which approximate the optimal configurations and we obtain tight upper bounds for the energy, which are better than the ones given by theoretical results. A thorough comparison of the results obtained by the three methods is given. We also investigate the behavior of the minimal partitions with respect to $p$. This allows us to see when partitions minimizing the 1-norm and the infinity-norm are different.
Calculus of variations and optimal control; optimization
Partial differential equations
Numerical analysis
129
163
10.4171/IFB/399
http://www.ems-ph.org/doi/10.4171/IFB/399
5
3
2018
2
Bubbles and droplets in a singular limit of the FitzHugh–Nagumo system
Chao-Nien
Chen
National Tsing Hua University, Hsinchu, Taiwan
Yung-Sze
Choi
University of Connecticut, Storrs, USA
Xiaofeng
Ren
George Washington University, Washington, USA
FitzHugh–Nagumo system, standing pulse, singular limit, nonlocal geometric variational problem, bubbles, droplet assemblies
The FitzHugh–Nagumo system gives rise to a geometric variational problem, when its parameters take values in a particular range. A stationary set of the variational problem satisfies an Euler–Lagrange equation that involves the curvature of the boundary of the set and a nonlocal term that inhibits unlimited growth and spreading. The nonlocal term is the solution of a nonhomogeneous modified Helmholtz equation with the characteristic function of the stationary set as the inhomogeneous term. Two types of stationary sets are studied: disc shaped stationary sets in the plane, termed bubbles, and unions of perturbed small discs, termed droplet assemblies, in bounded domains. A complete description of the existence and the stability of bubbles is established. Depending on the parameters, there may be zero, one, two, or even three bubbles. Droplet assemblies are constructed by a reduction argument. Each droplet in a stationary assembly is close in size and shape to the corresponding bubble in the plane. The locations of the droplets in the assembly are determined by the Green’s function of the modified Helmholtz equation.
Calculus of variations and optimal control; optimization
Special functions
Partial differential equations
Biology and other natural sciences
165
210
10.4171/IFB/400
http://www.ems-ph.org/doi/10.4171/IFB/400
7
9
2018
Uniform ball property and existence of optimal shapes for a wide class of geometric functionals
Jérémy
Dalphin
Université de Lorraine, Vandœuvre-Lès-Nancy, France
Shape optimization, uniform ball condition, geometric functionals, Helfrich, Willmore, curvature depending energies
In this article, we study shape optimization problems involving the geometry of surfaces (normal vector, principal curvatures). Given $\varepsilon > 0$ and a fixed non-empty large bounded open hold-all $B \subset \mathbb{R}^{n}$, $n \geqslant 2$, we consider a specific class $\mathcal{O}_{\varepsilon}(B)$ of open sets $\Omega \subset B$ satisfying a uniform $\varepsilon$-ball condition. First, we recall that this geometrical property $\Omega \in \mathcal{O}_{\varepsilon}(B)$ can be equivalently characterized in terms of $C^{1,1}$-regularity of the boundary $\partial \Omega \neq \emptyset$, and thus also in terms of positive reach and oriented distance function. Then, the main contribution of this paper is to prove the existence of a $C^{1,1}$-regular minimizer among $ \Omega \in \mathcal{O}_{\varepsilon}(B)$ for a general range of geometric functionals and constraints defined on the boundary $\partial \Omega$, involving the first- and second-order properties of surfaces, such as problems of the form: \[ \inf_{\Omega \in \mathcal{O}_{\varepsilon}(B)} \int_{\partial \Omega} \left( \begin{matrix} \\ \\ \end{matrix} j_{0} \left[ \mathbf{x},\mathbf{n}\left(\mathbf{x}\right) \right] ~+~ j_{1} \left[ \mathbf{x},\mathbf{n}\left(\mathbf{x}\right),H\left( \mathbf{x} \right)\right] ~+~ j_{2}\left[\mathbf{x},\mathbf{n}\left(\mathbf{x}\right),K\left(\mathbf{x}\right)\right] \begin{matrix} \\ \\ \end{matrix} \right) dA \left( \mathbf{x}\right), \] where $\mathbf{n}$, $H$, $K$ respectively denote the unit outward normal vector, the scalar mean curvature and the Gaussian curvature. We only assume continuity of $j_{0},j_{1},j_{2}$ with respect to the set of variables and convexity of $j_{1},j_{2}$ with respect to the last variable, but no growth condition on $j_{1},j_{2}$ are imposed here regarding the last variable. Finally, we give various applications in the modelling of red blood cells such as the Canham-Helfrich energy and the Willmore functional.
Calculus of variations and optimal control; optimization
Differential geometry
211
260
10.4171/IFB/401
http://www.ems-ph.org/doi/10.4171/IFB/401
7
9
2018
Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global s-minimal surfaces
Luca
Lombardini
Università degli Studi di Milano, Italy and Université de Picardie Jules Verne, Amiens, France
Nonlocal minimal surfaces, smooth approximation, existence theory, subgraphs
This article is divided into two parts. In the first part we show that a set $E$ has locally finite $s$-perimeter if and only if it can be approximated in an appropriate sense by smooth open sets. In the second part we prove some elementary properties of local and global $s$-minimal sets, such as existence and compactness. We also compare the two notions of minimizer (i.e., local and global), showing that in bounded open sets with Lipschitz boundary they coincide. Conversely, in general this is not true in unbounded open sets, where a global s-minimal set may fail to exist (we provide an example in the case of a cylinder $\Omega \times \mathbb R$).
Calculus of variations and optimal control; optimization
Partial differential equations
261
296
10.4171/IFB/402
http://www.ems-ph.org/doi/10.4171/IFB/402
7
9
2018
A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type
Marie
Henry
Université d’Aix-Marseille, Marseille, France
Danielle
Hilhorst
Université Paris-Sud, Orsay, France
Cyrill
Muratov
New Jersey Institute of Technology, Newark, USA
Pattern formation, multiscale analysis, singular perturbations, nonlinear dynamics
A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\mathbb R^N$, with $N \geq 2$. It is shown that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatio-temporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the considered class of reaction-diffusion equations.
Partial differential equations
297
336
10.4171/IFB/403
http://www.ems-ph.org/doi/10.4171/IFB/403
7
9
2018
3
Schwarz P surfaces and a non local perturbation of the perimeter
Matteo
Rizzi
Universidad de Chile, Santiago de Chile, Chile
Ohta–Kawasaki functional, Schwarz P surface, minimal surface, Lyapunov–Schmidt reduction
In the paper, we consider a small non local perturbation of the perimeter and we construct at least four critical points close to suitable translations of the Schwarz P surface with fixed volume.
Partial differential equations
337
352
10.4171/IFB/404
http://www.ems-ph.org/doi/10.4171/IFB/404
11
5
2018
Incompressible fluid problems on embedded surfaces: Modeling and variational formulations
Thomas
Jankuhn
RWTH Aachen, Germany
Maxim
Olshanskii
University of Houston, USA
Arnold
Reusken
RWTH Aachen, Germany
Fluids on surfaces, viscous material interface, fluidic membrane, Navier–Stokes equations on manifolds
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier–Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier–Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $\mathbb R^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.
Partial differential equations
Dynamical systems and ergodic theory
Fluid mechanics
353
377
10.4171/IFB/405
http://www.ems-ph.org/doi/10.4171/IFB/405
11
5
2018
A limit case in non-isotropic two-phase minimization problems driven by $p$-Laplacians
João Vítor
da Silva
Universidad de Buenos Aires, Argentina
Julio
Rossi
Universidad de Buenos Aires, Argentina
Free boundary problems, non-isotropic two-phase problems, $\infty$-Laplacian operator
In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of $p-$Laplacian type. The problem in its variational form is as follows: $$ \displaystyle \text{min} \left\{ \int\limits_{\Omega \cap \{v>0\}} \left(\frac{1}{p}|\nabla v|^p + \lambda_{+}^p+ f_{+}v \right)dx + \int\limits_{\Omega \cap \{v\leq 0\}} \left(\frac{1}{q}|\nabla v|^q + \lambda_{-}^q+ f_{-}v\right)dx \right\}. $$ Here we minimize among all admissible functions $v$ in an appropriate Sobolev space with a prescribed boundary datum $v=g$ on $\partial \Omega$. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where $p$ and $q$ go to infinity, obtaining a limiting free boundary problem governed by the $\infty$-Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.
Partial differential equations
379
406
10.4171/IFB/406
http://www.ems-ph.org/doi/10.4171/IFB/406
11
5
2018
Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone
Giovanni
Bellettini
Università di Siena, Italy and International Center for Theoretical Physics, Trieste, Italy
Maurizio
Paolini
Università Cattolica del Sacro Cuore, Brescia, Italy
Franco
Pasquarelli
Università Cattolica del Sacro Cuore, Brescia, Italy
Plateau problem, soap films, covering spaces
By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980’s, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.
Calculus of variations and optimal control; optimization
Manifolds and cell complexes
407
436
10.4171/IFB/407
http://www.ems-ph.org/doi/10.4171/IFB/407
11
5
2018
Two-dimensional steady supersonic exothermically reacting Euler flows with strong contact discontinuity over a Lipschitz wall
Wei
Xiang
City University of Hong Kong, Hong Kong
Yongqian
Zhang
Fudan University, Shanghai, China
Qin
Zhao
Fudan University, Shanghai, China
Supersonic flow, reacting Euler equations, Glimm scheme, fractional-step, Glimm functional, contact discontinuity, interface, stability, quasi-one-dimensional approximation
In this paper, we establish the global existence of supersonic entropy solutions with a strong contact discontinuity over a Lipschitz wall governed by the two-dimensional steady exothermically reacting Euler equations, when the total variation of both the initial data and slope of the Lipschitz wall is sufficiently small. Local and global estimates are developed and a modified Glimm-type functional is carefully designed. Next the validation of the quasi-one-dimensional approximation in the domain bounded by the wall and the strong contact discontinuity is rigorous justified by proving that the difference between the average of weak solution and the solution of quasi-one-dimensional system can be bounded by the square of the total variation of both the initial data and slope of the Lipschitz wall.
Partial differential equations
Fluid mechanics
437
481
10.4171/IFB/408
http://www.ems-ph.org/doi/10.4171/IFB/408
11
5
2018
4
Super-linear propagation for a general, local cane toads model
Christopher
Henderson
The University of Chicago, USA
Benoît
Perthame
Sorbonne Université, Université Paris Diderot, France
Panagiotis
Souganidis
The University of Chicago, USA
Reaction diffusion equations; long range, long time limits; motility; cane toads equation; mutation; spatial sorting, front propagation
We investigate a general, local version of the cane toads equation, models the spread of a population structured by unbounded motility. We use the thin-front limit approach of Evans and Souganidis in [Indiana Univ. Math. J., 1989] to obtain a characterization of the propagation in terms of both the linearized equation and a geometric front equation. In particular, we reduce the task of understanding the precise location of the front for a large class of equations to analyzing a much smaller class of Hamilton–Jacobi equations. We are then able to give an explicit formula for the front location in physical space. One advantage of our approach is that we do not use the explicit trajectories along which the population spreads, which was a basis of previous work. Our result allows for large oscillations in the motility.
Partial differential equations
Fluid mechanics
483
509
10.4171/IFB/409
http://www.ems-ph.org/doi/10.4171/IFB/409
12
13
2018
Thin obstacle problem: Estimates of the distance to the exact solution
Darya
Apushkinskaya
Universität des Saarlandes, Saarbrücken, Germany
Sergey
Repin
Russian Acadademy of Sciences, St. Petersburg, Russian Federation and Jyväskylä University, Finland
Thin obstacle, free boundary problems, variationals problems, estimates of the distance to the exact solution
We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., they are valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known.
Partial differential equations
Numerical analysis
511
531
10.4171/IFB/410
http://www.ems-ph.org/doi/10.4171/IFB/410
12
13
2018
Asymptotic stability of local Helfrich minimizers
Daniel
Lengeler
Universität Regensburg, Germany
Willmore energy, Canham–Helfrich energy, gradient flow, geometric flow,Willmore flow, Helfrich flow, Helfrich equation, Stokes system, linear elliptic system, fluid dynamics, biological membrane, lipid bilayer, well-posedness, stability
We show that local minimizers of the Canham–Helfrich energy are asymptotically stable with respect to a model for relaxational fluid vesicle dynamics that we already studied in previous papers ([13, 14]). The proof is based on a Łojasiewicz–Simon inequality.
Partial differential equations
Fluid mechanics
533
550
10.4171/IFB/411
http://www.ems-ph.org/doi/10.4171/IFB/411
12
13
2018
Approximation of minimal surfaces with free boundaries
Ulrich
Dierkes
Universität Duisburg-Essen, Germany
Tristan
Jenschke
Universität Duisburg-Essen, Germany
Paola
Pozzi
Universität Duisburg-Essen, Germany
Minimal surfaces, free boundary problem, finite element approximation, convergence
In this paper we develop a penalty method to approximate solutions of the free boundary problem for minimal surfaces. To this end we study the problem of finding minimizers of a functional $F_{\lambda}$ which is defined as the sum of the Dirichlet integral and an appropriate penalty term weighted by a parameter $\lambda$. We prove existence of a solution for $\lambda$ large enough as well as convergence to a solution of the free boundary problem as $\lambda$ tends to infinity. Additionally regularity at the boundary of these solutions is shown, which is crucial for deriving numerical error estimates. Since every solution is harmonic, the analysis is largely simplified by considering boundary values only and using harmonic extensions. In a subsequent paper we develop a fully discrete finite element procedure for approximating solutions to this problem and prove an error estimate which includes an order of convergence with respect to the grid size.
Differential geometry
Calculus of variations and optimal control; optimization
Numerical analysis
551
576
10.4171/IFB/412
http://www.ems-ph.org/doi/10.4171/IFB/412
12
13
2018
Free boundary regularity for a degenerate problem with right hand side
Raimundo
Leitão
Universidade Federal do Ceará, Fortaleza, Brazil
Gleydson
Ricarte
Universidade Federal do Ceará, Fortaleza, Brazil
Free boundary problems, degenerate elliptic operators, regularity theory
We consider a one-phase free boundary problem for $p$-Laplacian with non-zero right hand side. We use the approach present in [6] to prove that flat free boundaries and Lipschitz free boundaries are $C^{1, \gamma}$.
Partial differential equations
General
577
595
10.4171/IFB/413
http://www.ems-ph.org/doi/10.4171/IFB/413
12
13
2018